Maxwell’s Equations

“The theoretical prediction of electromagnetic waves was the work of the Scottish physicist James Clerk Maxwell (1831-879), who unified, in one magnificent theory, all the phenomena of electricity and magnetism.” [From Giancoli, p. 813.]

Gauss’s Law

ΦE =!E ⋅d!A

A"∫ =Qin A

ε0Charged particles create E- fields.# $%%% &%%%

Gauss’s Law for Magnetism

ΦB =!B ⋅d!A

A"∫ = 0Sourcesof B- fields(magnetic monopole)

do not exist.

# $%%% &%%%

Ampère-Maxwell’s Law

!B ⋅d!l

C"∫ = µ0I through Ctot

Ampère 's LawCurrents create B- fields.

# $%%% &%%%+ µ0ε0

dΦE

dtChanging E− fields

create B− fields.

# $% &%

Faraday’s Law

EMF = − dΦB

dtwith ΦB = − d

dt!B ⋅d!A

A∫( )or

!Eind ⋅d

!l

C"∫ = − d!Bdt

⋅d!A

A∫Changing B− fields create E− fields# $%%%% &%%%%

Remember ...

Coulomb constant k = 9 ×109 Nm2C2

ε0 =14π kPermittivity of free space ε0 = 8.85 ×10

−12 C2N ⋅m2

ε0 =14π k

Biot-Savart constant ′k = 10−7 NA2

µ0 =14π ′kPermeability of free space µ0 = 4π ×10−7 T⋅m

A

µ0 =14π ′k

Dr. Cereghetti - Physics 13 Maxwell’s Equations & Polarization – page 1

Lorentz Equation

The Lorentz equation gives the force on a particle that is produced by electric and magnetic fields.

!F = q

!E + !v ×

!B( )

Electromagnetic Waves

From Maxwell’s equations, one can also deduce the existence and the electromagnetic nature of light, or of any other electromagnetic wave.

“A magnetic field will be produced in empty space if there is a changing electric field. A changing magnetic field produces an electric field that is itself changing. This changing electric field will, in turn, produce a magnetic field, which will be changing, and so it too will produce a changing electric field; and so on. Maxwell found that the net result of these interacting changing fields was a wave of electric and magnetic fields that can propagate (travel) through space!” [From Giancoli, p. 813.]

Electromagnetic (EM) waves are transverse waves made of an electric field and a magnetic field perpendicular to each other that oscillate in magnitude perpendicular to the direction of motion of the wave. Because a moving charge generates time-changing electric and magnetic fields, an EM wave can be generated by an oscillating electric charge.

From Maxwell’s equations one can directly show that the speed of EM waves in empty space is given by:

c = EB= 1

ε0µ0= 3×108 m s

The relationship between wavelength λ and frequency f of an EM wave is:

c = λ f

B

B

B

E

E

E

x

z

y

B0

E0 wavelength λ velocity c

Dr. Cereghetti - Physics 13 Maxwell’s Equations & Polarization – page 2

In this handout,the Ampère-Maxwell's Law:

Changing E-fields create B-fields

From The Cartoon Guide to Physics by Larry Gonick & Art Huffman.

Dr. Cereghetti - Physics 13 Maxwell’s Equations & Polarization – page 3

Polarization & Polarization Filters

Note: In the following, we will deal with the electric interaction between EM-waves and matter. We will only consider the behavior of the electric field of an EM-wave. Although, for simplicity, we are leaving the magnetic field component of an EM-wave out of the picture, remember it is still there!

A single EM-wave can be linearly polarized, i.e. its electric field oscillates in a well defined plane of polarization (see picture on the side). When we consider EM-waves at the macroscopic level, like for example a ray of red light, what we are dealing with is actually a superposition of many individually polarized EM-waves. When the superimposed EM-waves building up the ray are polarized randomly, we say that that the ray is unpolarized.

Note that it is possible to have a polarized ray with all the superimposed EM-waves building up the ray sharing the same direction of motion, the same frequency, and the same polarization direction. See the following ...

“Some wave sources, such as lasers and radio antennas, emit polarized electromagnetic waves with a well-defined plane of polarization. By contrast, most natural sources of electromagnetic radiation are unpolarized. Each atom in the sun’s hot atmosphere emits light independently of all other atoms, as does each tiny piece of metal in the incandescent filament of a light bulb. An electromagnetic wave that you see or measure is a superposition of waves from each of these tiny emitters. Although the wave from each individual emitter is polarized, it is polarized in a random direction with respect to the waves from all its neighbors. The net result is what we call an unpolarized wave, a wave whose electric field oscillates randomly with all possible orientations.

A few natural sources are partially polarized, meaning that one direction of polarization is more prominent than others. The light of the sky at right angles to the sun is partially polarized, because of how the sun’s light scatters from air molecules to create skylight. Bees and other insects make use of this partial polarization to navigate. Light reflected from a flat, horizontal surface, such as a road or the surface of a lake, has a predominantly horizontal polarization. This is the rational for using polarizing sunglasses.

The most common way of artificially generating polarized visible light is to send unpolarized light through a polarizing filter: The first widely used polarizing filter was invented by Edwin Land in 1928, while he was still an undergraduate student. He developed an improved version, called Polaroid, in 1938. Polaroid (...) is a plastic sheet containing very long organic molecules known as polymers. The sheets are formed in such a way that the polymers are all aligned to form a grid, rather like the metal bars in a barbecue grill. The sheet is then chemically treated to make the polymer molecules somewhat conducting.

As light waves travels through Polaroid, the component of the electric field oscillating parallel to the polymer grid drives the conduction electrons up and down the molecules. The electrons absorb energy from the light wave, so the parallel component of E is absorbed in the filter. But the conduction electrons can’t oscillate perpendicular to the molecules, so the component of E perpendicular to the polymer grid passes through without absorption. Thus the light wave emerging from a polarizing filter is polarized perpendicular to the polymer grid.” [From R. Knight’s Physics for Scientists and Engineers, 1st ed., p. 1112.]

E

x

z

y

E0velocity c

planeofpolarization

Dr. Cereghetti - Physics 13 Maxwell’s Equations & Polarization – page 4

Effect of a Polaroid filter on light ...

The incoming light is unpolarized:

Unpolarized light with intensity I0 is incident on a polarizer like a Polaroid filter. After passing through the polarizer, the electric field of the light is parallel to the axis of the polarizer. The intensity of the transmitted light is:

I = I02

The vertical and horizontal components of unpolarized light have equal intensity. After passing through a polarizer, one of these components is absorbed reducing the intensity of the light to half.

The incoming light is polarized:

Polarized light with intensity I0 is incident on a Polaroid filter. After passing through the polarizer, the light changes its original polarization direction becoming polarized parallel to the axis of the Polaroid filter. The intensity of the transmitted light is:

I = I0 cos2θ Malus’s Law

where θ is the angle between the polarization direction of the incoming light and the axis of the Polaroid filter.

polarizer's axis

light'sdirection

unpolarizedlight

polarizedlight

I0

I02

polarizer

polarizer's axis

light'sdirection

polarizedlight

polarizedlight

I0

I = I0 cos2θ

polarizer

θ

Dr. Cereghetti - Physics 13 Maxwell’s Equations & Polarization – page 5

How to understand Malus’s Law ...The intensity of a wave is proportional to the square of the amplitude of the wave (that is, to the value of the electric field squared); therefore, we can express the intensity I0 of the incoming light and the intensity I of the filtered light (see picture on the side) as

I0 ∝E02 and I ∝E2 = E0

2 cos2θ

because for the filtered light only the component of the electric field along the polarizers’s axis remains. The ratio of I0 and I yields Malus’s Law:

I0I= E0

2

E02 cos2θ

→ I = I0 cos2θ

Polarizer and Analyzer:

“A Polaroid can be used as a polarizer to produce plane-polarized light from unpolarized light, since only the component of light parallel to the axis is transmitted. A Polaroid can also be used as an analyzer to determine (1) if light is polarized and (2) the plane of polarization. A Polaroid acting as an analyzer will pass the same amount of light independent of the orientation of its axis if the light is unpolarized; try rotating one lens of a pair of Polaroid sunglasses while looking through it at a lightbulb. If the light is polarized, however, when you rotate the Polaroid the transmitted light will be a maximum when the plane of polarization is parallel to the Polaroid’s axis, and a minimum when perpendicular to it. If you do this while looking at the sky, preferably at right angles to the sun’s direction, you will see that skylight is polarized. (Direct sunlight is unpolarized, but don’t look directly at the sun, even through a polarizer, for damage to the eye may occur.) If the light transmitted by an analyzer Polaroid falls to zero at one orientation, then the light is 100% plane-polarized. If it merely reaches a minimum, the light is partially polarized.” [From Giacoli’s Physics for Scientists and Engineers, p. 941.]

Two Polaroid filters in sequence:

When light goes sequentially through two Polaroid filters, the first filter is called the polarizer and the second filter the analyzer. When two Polaroid filters are crossed with their axes perpendicular to each other, unpolarized light can be entirely stopped:

Polaroid filter axisalong y-axis

y

x

polarization direction

of incoming lig

ht

E0 sinθis absorbed

E0 cosθpasses through

θ

E0incoming polarized light

polarizer's axis

light'sdirection

unpolarizedlight

polarizedlight

I0

I02

polarizer

analyzer's axis

analyzer

no light istransmitted

Dr. Cereghetti - Physics 13 Maxwell’s Equations & Polarization – page 6

Polarization by reflection:

“Another mean of producing polarized light from unpolarized light is by reflection. When light strikes a nonmetallic surface at any angle other than perpendicular, the reflected beam is polarized preferentially in the plane parallel to the surface (...). In other words, the component with polarization in the plane perpendicular to the surface is preferentially transmitted or absorbed. You can check this by rotating Polaroid Sunglasses while looking through them at a flat surface of a lake or road. Since most outdoor surfaces are horizontal, Polaroid sunglasses are made with their axes vertical to eliminate the more strongly reflected horizontal component, and thus reduce glare. People who go fishing wear Polaroids to eliminate reflected glare from the surface of a lake or stream and thus see beneath the water more clearly.” [From Giancoli’s Physics for Scientists and Engineers, p. 942.]

The amount of polarization in the reflected ray varies from no polarization at normal incidence to 100% polarization at an angle known as the polarizing angle θp. The polarizing angle depends on the index of refraction n1 and n2 of the two materials on either side of the boundary:

tanθp =n2n1

Brewster’s Law

If the incoming beam is traveling through air, n1=1, the polarizing angle is called Brewster’s angle.

θp

n1

n2

incomingunpolarized

light θpreflectedpolarized

light

refractedpartially polarized

light

θp

n1

n2

incomingin the page polarized

light

no reflectedlight

refractedin the page polarized

light

Dr. Cereghetti - Physics 13 Maxwell’s Equations & Polarization – page 7